To determine the completely factored form of [tex]\(d^4 - 81\)[/tex], we begin by recognizing that [tex]\(81\)[/tex] is a perfect square, as [tex]\(81 = 9^2\)[/tex]. Thus, we can rewrite the expression in a form that highlights this relationship:
[tex]\[ d^4 - 81 = d^4 - 9^2. \][/tex]
We notice that this fits the pattern of a difference of squares, [tex]\(a^2 - b^2\)[/tex], which factors into [tex]\((a - b)(a + b)\)[/tex]. Here, [tex]\(a = d^2\)[/tex] and [tex]\(b = 9\)[/tex]:
[tex]\[ d^4 - 81 = (d^2)^2 - 9^2 = (d^2 - 9)(d^2 + 9). \][/tex]
Next, we can factor the term [tex]\(d^2 - 9\)[/tex] further, as it is also a difference of squares. We apply the same pattern again, where [tex]\(d^2 - 9\)[/tex] can be written as:
[tex]\[ d^2 - 9 = (d - 3)(d + 3). \][/tex]
Thus, substituting back, we have:
[tex]\[ d^4 - 81 = (d^2 - 9)(d^2 + 9) = (d - 3)(d + 3)(d^2 + 9). \][/tex]
The term [tex]\(d^2 + 9\)[/tex] cannot be factored further as a real number expression since it does not fit the pattern for a difference of squares or any other recognizable factoring pattern for real numbers.
Hence, the completely factored form of [tex]\(d^4 - 81\)[/tex] is:
[tex]\[ (d - 3)(d + 3)(d^2 + 9). \][/tex]
